Tohoku Mathematical Journal

Counting pseudo-holomorphic discs in Calabi-Yau 3-holds

Kenji Fukaya

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In this paper we define an invariant of a pair of a 6 dimensional symplectic manifold with vanishing 1st Chern class and its relatively spin Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of bounding cochains (solutions of the $A_{\infty}$ version of the Maurer-Cartan equation of the filtered $A_{\infty}$ algebra associated to the Lagrangian submanifold). In the case when the Lagrangian submanifold is a rational homology sphere, it becomes a numerical invariant.

This invariant depends on the choice of almost complex structures. The way how it depends on the almost complex structures is described by a wall crossing formula which involves a moduli space of pseudo-holomorphic spheres.

Article information

Tohoku Math. J. (2), Volume 63, Number 4 (2011), 697-727.

First available in Project Euclid: 6 January 2012

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Zentralblatt MATH identifier

Primary: 57R17: Symplectic and contact topology
Secondary: 81T30: String and superstring theories; other extended objects (e.g., branes) [See also 83E30]

Symplectic geometry Lagrangian submanifold Floer homology Calabi-Yau manifold $A_{\infty}$ algebra superpotential mirror symmetry


Fukaya, Kenji. Counting pseudo-holomorphic discs in Calabi-Yau 3-holds. Tohoku Math. J. (2) 63 (2011), no. 4, 697--727. doi:10.2748/tmj/1325886287.

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